Volume 31, Issue 3

Given a partial action π of an inverse semigroup S on a ring 𝒜{\mathcal{A}}, one may construct its associated skew inverse semigroup ring 𝒜⋊πS{\mathcal{A}\rtimes_{\pi}S}. Our main result asserts that, when 𝒜{\mathcal{A}} is commutative, the ring 𝒜⋊πS{\mathcal{A}\rtimes_{\pi}S} is simple if, and only if, 𝒜{\mathcal{A}} is a maximal commutative subring of 𝒜⋊πS{\mathcal{A}\rtimes_{\pi}S} and 𝒜{\mathcal{A}} is S -simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra AR⁢(𝒢){A_{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid 𝒢{\mathcal{G}}.

We study Lie algebras of type I, that is, a Lie algebra 𝔤{\mathfrak{g}} where all the eigenvalues of the operator adX{\operatorname{ad}_{X}} are imaginary for all X∈𝔤{X\in\mathfrak{g}}. We prove that the Morse–Novikov cohomology of a Lie algebra of type I is trivial for any closed 1-form. We focus on locally conformal symplectic structures (LCS) on Lie algebras of type I. In particular, we show that for a Lie algebra of type I any LCS structure is of the first kind. We also exhibit lattices for some 6-dimensional Lie groups of type I admitting left invariant LCS structures in order to produce compact solvmanifolds equipped with an invariant LCS structure.

Let p⁢(⋅):ℝn→[0,1]{p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]} be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in L2⁢(ℝn){L^{2}({\mathbb{R}}^{n})}, with ω∈[0,π/2){\omega\in[0,\pi/2)}, which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space WHLp⁢(⋅)⁢(ℝn){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}, associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space WTp⁢(⋅)⁢(ℝ+n+1){\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})}, which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of WHLp⁢(⋅)⁢(ℝn){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}. As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform ∇⁡L-1/2{\nabla L^{-1/2}} is bounded from WHLp⁢(⋅)⁢(ℝn){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space WHp⁢(⋅)⁢(ℝn){\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})}. Moreover, when L is non-negative and self-adjoint with the kernels of {e-t⁢L}t>0{\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of WHLp⁢(⋅)⁢(ℝn){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.

We establish the mapping properties for some sublinear operators on weighted Hardy spaces with variable exponents by using extrapolation. In particular, we study the Calderón–Zygmund operators, the maximal Bochner–Riesz means, the intrinsic square functions and the Marcinkiewicz integrals on weighted Hardy spaces with variable exponents.

We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal . In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups ℤp{\mathbb{Z}_{p}} of p -adic integers. We extend Prodanov’s theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian. The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group ℚp⋊ℚp*{\mathbb{Q}_{p}\rtimes\mathbb{Q}_{p}^{*}} is hereditarily locally minimal, where ℚp*{\mathbb{Q}_{p}^{*}} is the multiplicative group of non-zero p -adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

We consider the generalized Jacobian J~{\widetilde{J}} of the modular curve X0⁢(N){X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the ℚ{\mathbb{Q}}-rational torsion points on J~{\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J~{\widetilde{J}} and its Eisenstein property.

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of (∞,1){(\infty,1)}-categories and ∞{\infty}-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

Let ϕ:G→H{\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair (G,ϕ-1⁢(L)){(G,\phi^{-1}(L))}, where L is a compact open subgroup of H . These results generalize the well-known properties of profinite completions to the locally compact setting.

We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of C1{C^{1}}-class (in view of two variables), but also allow for less regular functions, including discontinuous generators.

Let n≥2{n\geq 2} be an integer, 𝕊n{\mathbb{S}_{n}} the Lie algebra of divergence zero vector fields on an n -dimensional torus, and 𝒦n{\mathcal{K}_{n}} the Weyl algebra over the Laurent polynomial algebra An=ℂ⁢[x1±1,x2±1,…,xn±1]{A_{n}=\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}]}. For any 𝔰⁢𝔩n{\mathfrak{sl}_{n}}-module V and any module P over 𝒦n{\mathcal{K}_{n}}, we define an 𝕊n{\mathbb{S}_{n}}-module structure on the tensor product P⊗V{P\otimes V}. In this paper, necessary and sufficient conditions for the 𝕊n{\mathbb{S}_{n}}-modules P⊗V{P\otimes V} to be simple are given, and an isomorphism criterion for nonminuscule 𝕊n{\mathbb{S}_{n}}-modules is provided. More precisely, all nonminuscule 𝕊n{\mathbb{S}_{n}}-modules are simple, and pairwise nonisomorphic. For minuscule 𝕊n{\mathbb{S}_{n}}-modules, minimal and maximal submodules are concretely determined.

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function spt⁢(n){{\mathrm{spt}}(n)}. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function p⁢(n){p(n)} and spt⁢(n){{\mathrm{spt}}(n)}. Further, we strengthen one of the conjectures, and prove that for every ϵ>0{\epsilon>0} there is an effectively computable constant N⁢(ϵ)>0{N(\epsilon)>0} such that for all n≥N⁢(ϵ){n\geq N(\epsilon)}, we have 6π⁢n⁢p⁢(n)<spt⁢(n)<(6π+ϵ)⁢n⁢p⁢(n).\frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<{\mathrm{spt}}(n)<\bigg{(}\frac{\sqrt{6}}{% \pi}+\epsilon\bigg{)}\sqrt{n}\,p(n). Due to the conditional convergence of the Rademacher-type formula for spt⁢(n){{\mathrm{spt}}(n)}, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for p⁢(n){p(n)}. Instead, our approach relies on the fact that p⁢(n){p(n)} and spt⁢(n){{\mathrm{spt}}(n)} can be expressed as traces of singular moduli.

Let k be a field, let 𝔄1{\mathfrak{A}_{1}} be the k -algebra k⁢[x1±1,…,xs±1]{k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in x1,…,xs{x_{1},\dots,x_{s}}, and let 𝔄2{\mathfrak{A}_{2}} be the k -algebra k⁢[x,y]{k[x,y]} of polynomials in the commutative indeterminates x and y . Let σ1{\sigma_{1}} be the monomial k -automorphism of 𝔄1{\mathfrak{A}_{1}} given by A=(ai,j)∈G⁢Ls⁢(ℤ){A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and σ1⁢(xi)=∏j=1sxjai,j{\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}}, 1≤i≤s{1\leq i\leq s}, and let σ2∈Autk⁢(k⁢[x,y]){\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])}. Let Di{D_{i}}, 1≤i≤2{1\leq i\leq 2}, be the ring of fractions of the skew polynomial ring 𝔄i⁢[X;σi]{\mathfrak{A}_{i}[X;\sigma_{i}]}, and let Di∙{D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for D1{D_{1}}, and in general for D2{D_{2}}, we show that Di∙{D_{i}^{\bullet}}, 1≤i≤2{1\leq i\leq 2}, contains a free subgroup. If i=1{i=1} and s=2{s=2}, we show that a noncentral normal subgroup N of D1∙{D_{1}^{\bullet}} contains a free subgroup.

We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.

In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of Pk,sc{P_{k,{\mathrm{s}}}^{{\mathrm{c}}}}, Pk,s{P_{k,{\mathrm{s}}}}, Pk,wc{P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and Pk,w{P_{k,{\mathrm{w}}}}, the sets of all p∈[1,∞]{p\in[1,\infty]} for which the weak and strong type (p,p){(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators Mkc{M^{{\mathrm{c}}}_{k}} and Mk{M_{k}}, k≥1{k\geq 1}. For any fixed k we describe the necessary conditions that Pk,sc{P_{k,{\mathrm{s}}}^{{\mathrm{c}}}}, Pk,s{P_{k,{\mathrm{s}}}}, Pk,wc{P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and Pk,w{P_{k,{\mathrm{w}}}} must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k .

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝN{\mathbb{R}^{N}} (N∈ℕ+{N\in\mathbb{N}^{+}}), provided that the initial datum is sufficiently small. In addition, the Lp{L^{p}}-norm (1≤p≤∞{1\leq p\leq\infty}) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.